Results 211 to 220 of about 327,963 (245)
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Commutativity of rings with powers commuting on subsets
Mathematical Journal of Okayama University, 1997Let \(R\) denote a ring with 1; let \(w=w(X,Y)\) denote a word, possibly 1, in two noncommuting indeterminates; and let \(n\) be a positive integer. The elements \(x,y\in R\) are said to satisfy condition \(a(w,n)\) (resp. \(b(w,n)\)) if \(w(x,y)[x^n,y^n]=0\) (resp. \(w(x,y)((xy)^n-(yx)^n)=0\)). Define \(A\subseteq R\) to be a \(P\)-subset if for each \
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Journal of Algebra and Its Applications, 2014
A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.
Penk, Tomáš, Žemlička, Jan
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A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.
Penk, Tomáš, Žemlička, Jan
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Mathematics of the USSR-Sbornik, 1976
This paper deals with one-dimensional (commutative) rings without nilpotent elements such that every ideal is generated by three elements. It is shown that in such rings the square of every ideal is invertible, i.e. divides its multiplier ring. In addition, every ideal is distinguished, in the sense that on localization at any maximal ideal it becomes ...
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This paper deals with one-dimensional (commutative) rings without nilpotent elements such that every ideal is generated by three elements. It is shown that in such rings the square of every ideal is invertible, i.e. divides its multiplier ring. In addition, every ideal is distinguished, in the sense that on localization at any maximal ideal it becomes ...
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A Commutativity Condition for Rings
Canadian Journal of Mathematics, 1976The object of this paper is to prove the following theorem, a special case of which was previously explored in [1].THEOREM. Let R be any associative ring with the property that(†) for each x,y ∊ R, there exist integers m,n ≧ I for which xy = ymxn.
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On the commutativity of non-associative rings
Publicationes Mathematicae Debrecen, 2022\textit{E. C. Johnsen}, \textit{D. L. Outcalt}, and \textit{A. Yaqub} proved that a (not necessarily associative) ring with identity satisfying \((xy)^2 = x^2y^2\) for all \(x,y\) is commutative [Am. Math. Mon. 75, 288--289 (1968; Zbl 0162.33602)].
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Canadian Journal of Mathematics, 1982
Throughout this paper R will be a commutative ring with 1. The purpose of this paper is to provide two new characterizations of coherent rings. The first of these characterizations shows that the class of coherent rings is precisely the class of rings for which certain duality homomorphisms are isomorphisms.
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Throughout this paper R will be a commutative ring with 1. The purpose of this paper is to provide two new characterizations of coherent rings. The first of these characterizations shows that the class of coherent rings is precisely the class of rings for which certain duality homomorphisms are isomorphisms.
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On Commuting Rings Of Endomorphisms
Canadian Journal of Mathematics, 1956Various problems concerning the general theory of centralizers of modules which are not assumed to be completely reducible have been discussed by Fitting (3), Brauer (2), and Nakayama. In this paper we present a new approach to some of these questions, which has its origin in Weyl's discussion (15) of the centralizer of a finite group of collineations.
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Commutativity of rings with constraints on commutators
2000This paper studies commutativity of rings \(R\) satisfying polynomial identities of the form\break \(x^t[x^n,y]y^r=[x,y^m]y^s\) and three similar forms, where \(n,m,r,s,t\) are suitably-chosen nonnegative integers. Whether the theorems are correct as stated is not clear, but for some \((n,m,r,s,t)\) the proofs given do not work.
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On the Commutativity of a Ring with Identity
Canadian Mathematical Bulletin, 1984AbstractLet R be a ring with identity. R satisfies one of the following properties for all x, y ∈ R:(I)xynxmy = xm+1yn+1 and mnm! n! x≠0 except x = 0;(II)xynxm = xm + 1yn + 1 and mm! n! x≠0 except x = 0;(III)xmyn = ynxm and m! n! x≠0 except x = 0;(IV)(xpyQ)n = xpnyqn for n = k, k + 1 and N(p, q, k) x≠0 except x = 0, where N(p, q, k) is a definite ...
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A Condition for the Commutativity of Rings
Canadian Journal of Mathematics, 1957A well-known theorem of Jacobson (1) asserts that if every element a of a ring A satisfies a relation an(a) = a where n(a) > 1 is an integer, then A is a commutative ring.
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